458678 Model Predictive Control for Optimal Zone Tracking
In zone tracking, the stage cost is zero if the state is in the target set and positive otherwise. One such function that satisfies this property is the distance function from a point to a set. Under the stated assumptions, a Lyapunov function for the closed-loop system is derived to prove that the target set is asymptotically stable. A key practical consideration is that the solution to the optimal control problem when tracking zones is often not unique. There may exist a set of input sequences that can maintain the state within the target set. In this case, it is desirable to achieve secondary control objectives. To this end, other terms can be added to the objective function to minimize economic costs, input usage, or input movement to select the most desirable state trajectory within the target set for the given process conditions. A general framework is proposed to achieve these goals. After solving the zone tracking problem to determine the set of stabilizing input trajectories, a second optimization problem can be solved to determine the unique input trajectory from the stabilizing set that optimizes secondary control objectives. While this method requires solving two optimization problems at each MPC step, the procedure is simplified to solving a single optimization problem using exact penalties. Exact penalties on the deviation of the state from the target set can be used to ensure that the solution does not leave the target set if feasible. The resulting solution leads to stabilizing behavior if the penalty coefficient is chosen to be large enough.
The proposed formulation is applied to two example problems. First, a buffer tank between upstream and downstream process units is considered. The goal of the buffer tank is to prevent the upstream flow disturbance from propagating downstream by allowing the height of material in the tank to vary. While the height is allowed to vary, it should be kept within bounds to prevent emptying or overflowing. Through a simulation study, we show that the proposed MPC formulation accomplishes these goals. The controller changes the outflow of the tank to get the height within the bounds. When the height is within bounds, the controller keeps the outflow of the tank constant as long as the height stays in the target zone.
The second example problem considered is indoor air temperature control. The air temperature in commercial buildings is not required to be at a fixed setpoint. The air temperature should be maintained between upper and lower bounds to keep the occupants in the building comfortable. If the temperature is outside of these bounds, the controller should take action to drive the temperature to the comfort region. When the temperature is inside the bounds, the controller should minimize the amount of energy used by the heating, ventilation, and air conditioning (HVAC) system. We demonstrate that the proposed MPC formulation tracks the comfort region by rejecting the ambient disturbance. When the temperature is inside the comfort region, the controller avoids using the HVAC system to cool if possible by taking advantage of the disturbance while keeping occupants satisfied.
Qin, S. J., & Badgwell, T. A. (2003). A survey of industrial model predictive control technology. Control Engineering Practice, 11 (7), 733-764.
Rawlings, J. B., & Mayne, D. Q. (2009). Model Predictive Control: Theory and Design. Madison, WI: Nob Hill Publishing.